SETS
Set-builder notation:
notation for describing a set using variables
Examples: C = {x|x is
a natural number smaller than 4}
D = {3,4}
Membership: the
symbol Є means “is an element of.”
1 Є C, 4 is not a Є of C
Union: A U B = {x|x Є
A or x Є B}
C U D = {1,2,3,4}
Intersection: A n B =
{x|x Є A and x Є B}
C n D = {3}
Subset: A is a subset
of B if every element of A is also an element of B. The symbol c means “is a subset of.”
Ф c for any set A.
{1,2} = c C
Ф c C, Ф c D
REAL NUMBERS
Rational numbers: Q =
{a/b} a and b are integers with b ≠ 0}
3/2, 5, -6 0, 0.25252525 . . .
Irrational numbers: I
= {x|x is a real number that is not rational}
√2, √3, ∏, 0.1515515551 . . .
Intervals of real numbers:
An interval of real numbers is the set of real numbers that lie between
two real numbers, which are called the endpoints of the interval. We may use -∞ or ∞ as endpoints.
The real numbers between 3 and 4: (3,4)
The real numbers greater than or equal to 6: [6, ∞)
OPERATIONS WITH REAL NUMBERS
Absolute value: |a| =
{a if a is positive or zero; -a if a is negative}
|6| = 6, |0| = 0
|-6| = 6
Addition and subtraction:
To find the sum of two numbers with the same sign, add their absolute
values. The sum has the same sign as the
original numbers.
-2 + (-7) = -9
To find the sum of two numbers with unlike signs, subtract
their absolute values. The sum is
positive if the number with the larger absolute value is positive. The sum is negative if the number with the
larger absolute value is negative.
-6 + 9 = 3
-9 + 6 = -3
Subtraction: a – b =
a + (-b)—(Change the sign and add.)
4 – 7 = 4 + (-7) = -3
5 – (-3) = 5 + 3 = 8
Multiplication and division:
To find the product or quotient of two numbers, multiply or divide their
absolute values: Same signs ↔ positive result. Opposite signs ↔ negative result.
(-4)(-2) = 8, (-4)(2) = -8
-8 divide (-2) = 4, -8 divide 2 = -4
Exponential expressions:
In the expression an, a is the base and n is the exponent.
23 = 2 · 2 · 2 = 8
Square roots: If a2
= b, then is a square root of b. If a ≥
0 and a2 = b, then √b = a.
Both 3 and -3 are
square roots of 9. Because 3 ≥ 0. √9 =
3.
Order of operations:
In an expression without parentheses or absolute value:
1.
Evaluate exponential expressions.
2.
Perform multiplication and division.
3.
Perform addition and subtraction.
With parentheses or absolute value:
1.
First evaluate within each set of parentheses or
absolute value, using the preceding
order.
(2 + 4)(5 – 9) = -24
3 + 4|2 - 3| = 7
PROPERTIES OF THE REAL NUMB ERS
Commutative property of addition/multiplication: For any real numbers a, b, and c:
a + b = b + c
ab = ba
3 + 7 = 7 + 3
4 · 3 = 3 · 4
Associative property of addition/multiplication: (a + b) + c = a + (b + c); (ab)c = a(bc)
(1 + 3) + 5 = 1 + (3 + 5)
(3 · 5)7 = 3(5 · 7)
Distributive property:
a(b + c) = ab + ac
3(4 + x) = 12 + 3x
5x - 10 = 5(x – 2)
Additive identity property:
a + 0 = 0 + a = a
6 + 0 = 0 + 6 = 6
Multiplicative identity property: 1 · a = a · 1 = a
1 · 6 = 6 · 1 = 6
Additive inverse property:
a + (-a) = -a + a = 0
8 + (-8) = -8 + 8 = 0
Multiplicative inverse property: a · 1/a = 1/a · a = a for a ≠ 0
8 · 1/8 = 1, -2(-1/2) = 1
Multiplication property of zero: 0 · a = a · 0 = 0
9 · 0 = 0
(0)(-4) = 0
ALGEBRAIC CONCEPTS
Algebraic expressions:
Any meaningful combination of numbers, variables, and operations
X2 + y2, -5abc
Term: An expression
containing a number or the product of a number and one or more variables raised
to powers.
3x2, -7x2y, 8
Like terms: Terms
with identical variable parts
4bc – 8bc = -4bc
CHAPTER 2
EQUATIONS
Solution
set: the set of all numbers that satisfy
an equation (or inequality)
X + 2 = 6 has solution set {4}
Equivalent equations:
Equations with the same solution set
2x + 1 = 5
2x = 4
Properties of equality:
We may perform the same operation ( +, -, ·, divide) with the same real
number on each side of an equation without changing the solution set (excluding
multiplication and division by 0).
x = 4
x + 1 = 5
x – 1 = 3
2x = 8
x/2 = 2
Identity: An equation
that is satisfied by every number for which both sides are defined
x + x = 2x
Conditional equation:
An equation whose solution set contains at least one real number but is
not an identity
5x – 10 = 0
Inconsistent equation:
An equation whose solution set is Ф
x = x + 1
Linear equation in one variable: An equation of the form ax = b with a ≠ 0 or
an equation that can be rewritten in this form
3x + 8 = 0
5x – 1 = 2x – 9
Strategy for solving a linear equality
1.
If fractions are present, multiply each side by
the LCD to eliminate the fractions.
2.
Use the distributive property to remove
parentheses.
3.
Combine any like terms.
4.
Use the addition property of equality to get all
variables on one side and numbers on the other side.
5.
Use the multiplication property of equality to
get a single variable on one side.
6.
Check by replacing the variable in the original
equation with your solution.
Strategy for solving word problems
1.
Read the problem until you understand the
problem.
2.
If possible, draw a diagram to illustrate the
problem.
3.
Choose a variable and write down what it
represents.
4.
Represent any other unknowns in terms of that
variable.
5.
Write an equation that models the situation.
6.
Solve the equation.
7.
Be sure that your solution answers the question
posed in the original problem.
8.
Check your answer by using it to solve the
original problem (not the equation).
INEQUALITIES
Linear inequality in one variable: Any inequality of the form ax < b with a ≠
0 or an inequality that can be rewritten in this form. In place of < we can use ≤, > or ≥.
2x + 9 < 0
X – 2 ≥ 7
-3x – 1 ≥ 2x + 5
Properties of inequality:
We may perform the same operation (+, -, ·, divide) on each side of an
inequality just as we do in solving equations, with one exception: When multiplying or dividing by a negative
number, the inequality symbol is reversed.
-3x > 6
X < -2
Trichotomy property:
For any two real numbers a and b, exactly one of the following
statements is true:
a < b, a = b, or a > b
If w is not greater than 7, then w ≤ 7.
Compound inequality:
Two simple inequalities connected with the word “and” or “or”
And corresponds to intersection.
Or corresponds to union.
X > 1 and x < 5
X > 3 or x < 1
ABSOLUTE VALUE
Basic absolute value equations
Absolute Value Equation:
|x| = k (k > 0); |x| = 0; |x| = k (k < 0)
Equivalent Equation:
x = k or x = -k; x = 0
Solution Set: {k,
-k}; {0}; Ф
Basic absolute value inequalities (k > 0)
Absolute Value Equality:
|x| > k; |x| ≥ k; |x| < k; |x| ≤ k
Equivalent Inequality:
x > k or x < k; x ≥ k or x ≤ -k; -k < x < k; -k ≤ x ≤ k
Solution Set: (-∞,
-k) U (k, ∞); (-∞, -k) U [k, ∞); (-k, k); [-k, k]
CHAPTER 3
RECTANGULAR COORDINATE SYSTEM
X-intercept: The
point where a non-horizontal line intersects the x-axis
Y-intercept: the
point where a non-vertical line intersects the y-axis
For the line 2x + y = 6, the x-intercept is (3, 0) and the
y-intercept is (0, 6).
SLOPE
Slope of a line:
Slope = change in y-coordinate/change in x-coordinate = rise/run
Slope using coordinates:
Slope of line through (x1, y1) and (x2,
y2)is m = y2 – y1/x2 – x1,
provided that x2 –x1 ≠0.
If (x1, y2) = (4, -2) and (x2,
y2) = (3, -6), then m = -6 – (-2)/3 – 4 = 4.
Types of slope:
1.
Positive slope:
increases
2.
Negative slope:
decreases
3.
Zero slope:
horizontal
4.
Undefined slope:
vertical
Perpendicular lines:
The slope of one line is the opposite of the reciprocal of the slope of
the other line.
The lines y = 1/3x + 5 and y = 3x – 9 are perpendicular.
Parallel lines:
Non-vertical parallel lines have equal slopes.
The lines y = 2x -3 and y = 2x + 7 are parallel.
FORMS OF LINEAR EQUATIONS
Point-slope form: y –
y1 = m(x – x1); (x1, y1) is a point
on the line, and m is the slope.
Line through (5, -3) with slope 2: y + 3 = 2(x – 5)
Slope-intercept form:
y = mx + b; m is the slope, (0, b) is the y-intercept
Line through (0,-3) with slope 2: y = 2x -3
Standard form: Ax +
By = C; A and B are not both 0.
3x – 2y = 12
Vertical line: x = k,
where k is any real number. Slope is
undefined for vertical lines.
x = 5
Horizontal line: y = k,
where k is any real number. Slope is
zero for horizontal lines.
y = -2
GRAPHING LINEAR EQUATIONS
Point-plotting:
Arbitrarily select some points that satisfy the equation, and draw a
line through them.
For y = 2x + 1, draw a line through (0, 1), (1, 3), and (2,
5).
Intercepts: Find the
x- and y-intercepts (provided that they are not the origin), and draw a line
through them.
For x + y = 4 the intercepts are (0, 4) and (4, 0).
y-intercept and slope:
start at the y-intercept and use the slope to locate a second point,
then draw a line through the two points.
For y = 3x – 2 start at (0, -2), rise 3 and run 1 to get to
(1, 1), two points.
LINEAR INEQUALITIES
Linear inequality: Ax
+ By ≤ C, where A and B are not both zero.
The symbols <, >, and ≥ are also used.
2x – 3y ≤ 7
x – y > 6
Graphing linear inequalities: Solve for y, then graph the line y = mx + b.
y > mx + b is the region above the line.
y < mx + b is the region below the line.
For inequalities without y, graph x = k.
x > k is the region to the right of x = k.
x < k is the region to the left of x = k.
Graph of y = x + 2 is a line.
y > x + 2 is above y = x + 2.
y < x + 2 is below y = x + 2.
The graph of x > 5 is to the right of the vertical line x
= 5, and the graph of x < 5 is to the left of x = 5.
Test points: A linear
inequality may also be graphed by graphing the corresponding line and then
testing a point to determine which region satisfies the inequality.
COMPOUND INEQUALITIES
In one variable (from Section 2.5)
Two simple inequalities in one variable connected with the
word “and” or “or”
The solution set for an “and” inequality is the intersection
of the solution sets.
The solution set for an “or” inequality is the union of the
solution sets.
In two variables
Two simple inequalities in two variables connected with the
word “and” or “or”
The solution set for an “and” inequality is the intersection
of the solution sets.
The solution set for an “or” inequality is the union of the
solution sets.
Note that the graph of x > 1 (an inequality containing
only one variable) in the rectangular coordinate system is the region to the
right of the vertical line x = 1.
RELATIONS AND FUNCTIONS
Relation: Any set of
ordered pairs of real numbers
{(1,2), (1,3)}
Function: A relation
in which no two ordered pairs have the same first coordinate and different
second coordinates.
If y is a function of x, then y is uniquely determined by
x. A function may be defined by a table,
a listing of ordered pairs, or an equation.
{(1,2), (3,5), (4,5)}
Domain: The set of
first coordinates of the ordered pairs.
Function: y = x2,
Domain: (-∞, ∞)
Range: The set of
second coordinates of the ordered pairs.
Function: y = x2,
Range: [0, ∞)
Function notation: if
y is a function of x, the expression f(x) is used in place of y.
Y = 2x + 3
f(x) = 2x + 3
Vertical-line test:
If a graph can be crossed more than once by a vertical line, then it is
not the graph of a function.
Linear function: A
function of the form f(x) = mx + b with m ≠ 0
f(x) = 3x + 7
f(x) = -2x + 5
Constant function: A
function of the form f(x) = b, where b is a real number
f(x) = 2
CHAPTER 4
SYSTEMS OF LINEAR EQUATIONS
Methods for solving systems in two variables: Graphing:
sketch the graphs to see the solution.
The graphs or y = x – 1 and x + y = 1 and x + y = 3
intersect at (2, 1)
Substitution: Solve
one equation for one variable in terms of the order, then substitute into the
other equation.
x + (x + 1) = 3
Addition: Multiply
each equation as necessary to eliminate a variable upon addition of the
equations.
-x + y = -1
x + y = 3
2y = 2
Types of linear systems in two variables:
Independent: One
point in solution set. The lines
intersect at one point.
y = x – 5; y = 2x + 3
Dependent: Infinite
solution set. The lines are the same.
2x + 3y = 4; 4x + 6y = 8
Inconsistent: Empty
solution set. The lines are parallel.
2x + y = 1; 2x + y = 5
Linear equation in three variables: Ax + By + Cz = D. In a three-dimensional coordinate system the
graph is a plane.
2x – y + 3z = 5
Linear systems in three variables: Use substitution or addition to eliminate
variables in the system. The solution
set may be a single point, the empty set, or an infinite set of points.
x + y – z = 3
2x – y + z = 2
x – y – 4z = 14
MATRICES AND DETERMINANTS
Matrix: A rectangular
array of real numbers. An m x n matrix
has m rows and n columns.
[1 -3] · [1 0 1]
[2 5] · [2 1 4]
Augmented matrix: The
matrix of coefficients and constants from a system of linear equations.
x – 3y = -7
2x + 5y = 19
Augmented matrix:
[1 -3|-7]
[2 5|19]
Gauss-Jordan elimination method: Use the row operations to get ones on the
diagonal and zeros elsewhere for the coefficients in the augmented matrix.
[1 0|2]
[0 1|3]
x = 2 and y = 3
Determinant: A real
number corresponding to a square matrix.
Determinant of a 2 x 2 matrix: |a1 b1| = a1b2
– a2b1
|a2 b2|
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